Flexural Strengthening of Beams and Slabs - Strengthening Design of Reinforced Concrete with FRP - page 80

5.3 STRENGTH REDUCTION FACTORS
The strength reduction factor of a strengthened section is given by Equation (10.5)
of ACI 440.2R-08 based on the requirements of ACI 318-05 (2005). This factor
ϕ is a function of the net tensile strain in the most extreme tension steel layer at
nominal strength:

0.9 for ε≥ 0.005

t

( )

(5.2)

0.25
0.005

ε−ε
−ε

t sy

φ=

0.65 + for ε<ε< 0.005

sy t

sy

0.65 for ε≤ε

t sy

The above values are set to ϕ = 0.9 for a tension-controlled failure (ductile) and to
ϕ = 0.65 for a compression-controlled failure (brittle without yielding), while ϕ is
determined from the linear interpolation between the two extremes in the transition
zone, as seen in Figure 5.1.
For Grade 60 reinforcement, ε= ≈
sy

60 ksi
29,000 ksi

0.002 . Thus for φ= 0.9,

d t

≤ 0.375 ,

d
c t

and for φ= + − . In addition to

the factor ϕ, another strength reduction fact or for FRP (Ψ f ) is multiplied by the
flexural contribution of FRP reinforcement M nf

φ= 0.65,

d t

≥ 0.6 . In the transition, 0.65 0.25

3

( ), as shown in Equation (10.13) of

the ACI 440.2R-08:

− β

c

− β

c

1 1

MAfd = +Ψ Af d (5.3)

n s s f f fe f

2 2

This Ψ f factor is taken as 0.85 based on the reliability analysis of the experimentally
calibrated statistical values to mainly account for the less predictable failure mode of
delamination of FRP reinforcement (Okeil, Bingol, and Alkhrdaji 2007).

φ factor

0.90

0.65

Compression
Controlled
Zone

Transition Zone Tension Controlled Zone

Net Steel Strain at Ultimate

ε sy

0.005

FIGURE 5.1 Variation of ϕ factor as a function of the extreme net steel strain.